1. Prove that
is not a rational number.
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Let be a rational number.
, where q and p are co-primes with no common factor other than 1
Squaring on both sides:
We get:
---(1)
Thus;
||
And so,
|| p
Now, we know that:
p =
Squaring on both sides:
Put (1):
Crossing the same terms, we get:
Thus;
||
And so,
|| q
But p and q were co-primes.
This contradicts our assumption. As, p and q have common factor as .
Thus, is an irrational number.
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